Table Of ContentUniversal Tree Source Coding Using
Grammar-Based Compression∗
Danny Hucke and Markus Lohrey
Universit¨at Siegen, Germany
7
1
0
2 February 1, 2017
n
a
J
1
2 Abstract
] We apply so-called tree straight-line programs to the problem of
T
universal source coding for binary trees. We derive an upper bound
I onthemaximalpointwiseredundancy(orworst-caseredundancy)that
.
s
improve previous bounds on the averagecase redundancy obtained by
c
[ Zhang,Yang,andKiefferusingdirectedacyclicgraphs. Usingthis, we
obtain universal codes for new classes of tree sources.
1
v
5
1 Introduction
8
7
8 Universal source coding for finite sequences over a finite alphabet Σ (i.e.,
0
strings over Σ) is a well-established topic of information theory. Its goal is
.
1
to find prefix-free lossless codes that are universal (or optimal) for classes
0
7 of information sources. In a series of papers, Cosman, Kieffer, Nelson, and
1 Yang developed grammar-based codes that are universal for the class of
:
v finite state sources [8, 9, 10, 16]. Grammar-based compression works in
Xi two steps: In a first step, from a given input string w ∈ Σ∗ a context-free
grammar G that produces only the string w is computed. Context-free
r w
a
grammars that produce exactly one string are also known as straight-line
programs, brieflySLPs,andarecurrentlyanactive topicintextcompression
and algorithmics on compressed texts, see [12] for a survey. In a second
step, the SLP G is encoded by a binary string B(G ). There exist several
w w
algorithms that compute from a given inputstring w of length n an SLP G
w
of size O(n/logn) (the size of an SLP is the total number of symbols in all
right-hand sides of the grammar) [8]; the best known example is probably
the LZ78 algorithm [18]. By combining any of these algorithms with the
∗This work has been supported by the DFG research project LO 748/10-1 (QUANT-
KOMP).
1
binary encoder B for SLPs from [8], one obtains a grammar-based encoder
E : Σ∗ → {0,1}∗, whose worst case redundancy for input strings of length
n is bounded by O(loglogn/logn) for every finite state information source
over the alphabet Σ. Here, the worst case redundancy for strings of length
n is defined as
max n−1·(|E(w)|+log P(w)),
2
w∈Σn,P(w)>0
where P(w) is the probability that the finite state information source emits
w. Thus, the worst case redundancy measures the maximal additive devia-
tion of the code length from the self information, normalized by the length
of the source string.
Over the last few years, we have seen increasing efforts aiming to extend
universal source coding to structured data like trees [11, 14, 17] and graphs
[2, 7]. In this paper, we are concerned with universal source coding for
trees. In their recent paper [17], Kieffer, Yang, and Zhang started to extend
their work on grammar-based source coding from strings to binary trees.
For this, they first represent the input tree t by its minimal directed acyclic
graphD (theminimalDAGoft). Thisisthedirectedacyclicgraphobtained
t
by removing multiple occurrences of the same subtree from t. In a second
step, the minimal DAG D is encoded by a binary string B(D ); this step is
t t
similar to the binary coding of SLPs from [8]. Combining both steps yields
a tree encoder E : T → {0,1}∗, where T denotes the set of all binary
dag
trees. In order to define universality of such a tree encoder, the classical
notion of an information source on finite sequences is replaced in [17] by
the notion of a structured tree source. The precise definition can be found
in Section 2.2; for the moment the reader can think about a collection of
probabilitydistributions(Pn)n∈N,whereeveryPn isadistributiononafinite
non-emptysubsetF , andthesesetspartitionT. Themaincases considered
n
in [17] as well as in this paper are: (i) F is the set of all binary trees with
n
n leaves (leaf-centric sources) and (ii) F is the set of all binary trees of
n
depth n (depth-centric sources). Then, the authors of [17] introduce two
propertiesonbinarytreesources: (i)thedominationproperty(seeSection3,
where it is called the weak domination property) and (ii) the representation
ratio negligibility property. The latter states that P (t)·|D |/|t| (the
t∈Fn n t
average compression ratio achieved by the minimaPl DAG) converges to zero
for n → ∞, where the size |t| of the binary tree is defined as its number of
leaves. Thetechnical main resultof [17]states that for every structuredtree
source (Pn)n∈N satisfying the domination property and the representation
ratio negligibility property the average case redundancy
|t|−1·(|E (t)|+log P (t))·P (t)
dag 2 n n
t∈FnX,Pn(t)>0
converges to zero for n → ∞. Finally, two classes of tree sources having the
domination property and the representation ratio negligibility property are
2
presentedin[17]. Oneisaclassofleafcentricsources,theotheroneisaclass
ofdepthcentricsources. Bothsourceshavethepropertythateverytreewith
a non-zero probability is balanced in a certain sense, the precise definitions
can be found in Section 3.3 and Section 3.4. As a first contribution, we
show that for these sources not only the average case redundancy but also
the worst case redundancy
max |t|−1·(|E (t)|+log P (t)) (1)
dag 2 n
t∈Fn,Pn(t)>0
converges to zero for n → ∞. More precisely, we show that (1) is bounded
by O(loglogn/logn) (respectively, O((loglogn)2/logn)) for the presented
class of leaf-centric tree sources (respectively, depth-centric tree sources).
To prove this, we use results from [5, 6] according to which the minimal
DAG of a suitably balanced binary tree of size n is bounded by O(n/logn),
respectively O(n·loglogn/logn).
Our second main contribution is the application of tree straight-line pro-
grams, briefly TSLPs, for universal tree coding. A TSLP is a context-free
tree grammar that produces exactly one tree, see Section 2.3 for the pre-
cise definition and [13] for a survey. TSLPs can be viewed as the proper
generalization of SLPs for trees. Whereas DAGs only have the ability to
share repeated subtrees of a tree, TSLPs can also share repeated tree pat-
terns with a hole (so-called contexts). In [5], the authors presented a linear
time algorithm that computes for a given binary tree t of size n a TSLP G
t
of size O(n/logn). This shows the main advantage of TSLPs over DAGs:
There exist trees of any size n for which the minimal DAG has size n as
well. In Section 4.2 we define a binary encoding B of TSLPs similar to
the ones for SLPs [8] and DAGs [17]. We then consider the combined tree
encoder E : T → {0,1}∗ with E (t) = B(G ), and prove that its worst
tslp tslp t
case redundancy (that is defined as in (1) with E replaced by E ) is
dag tslp
bounded by O(loglogn/logn) for every structured tree source that satisfies
the strong domination property defined in Section 4.3. The strong domina-
tion property is a strengthening of the domination property from [17], and
this is what we have to pay extra for our TSLP-based encoding in contrast
to the DAG-based encodingfrom [17]. Onthe other hand, our approach has
two main advantages over [17]:
• The representation ratio negligibility property from [17] is no longer
needed.
• We get bounds on the worst case redundancy instead of the average
case redundancy.
Both advantages are based on the fact that the grammar-based compressor
from [5] computes a TSLP of worst case size O(n/logn) for a binary tree of
size n.
3
Finally, we present a class of leaf-centric sources (Section 3.3) as well as
a class of depth-centric sources (Section 3.4) having the strong domination
property. These classes are orthogonal to the classes considered in [17].
2 Preliminaries
In this section, we introduce some basic definitions concerning information
theory (Section 2.1), binary trees (Section 2.2) and tree straight-line pro-
grams (Section 2.3). The latter are our key formalism for the compression
of binary trees.
WithNwedenotethenaturalnumbersincluding0. Weusethestandard
O-notation and if b is a constant, then we just write O(logn) for O(log n).
b
For the unit interval {r ∈ R | 0≤ r ≤ 1} we write [0,1].
2.1 Empirical distributions and empirical entropy
Let a = (a ,a ,...,a ) be a tuple of elements that are from some (not
1 2 n
necessarilyfinite)setA. Theempirical distributionp :{a ,a ,...,a } → R
a 1 2 n
of a is defined by
|{i |1 ≤ i ≤ n, a = a}|
i
p (a) = .
a
n
We use this definition also for words over some alphabet by identifying a
word w = a a ···a with the tuple (a ,a ,...,a ). The unnormalized
1 2 n 1 2 n
empirical entropy of a is
n
H(a) = − logp (a ).
a i
Xi=1
A well-known generalization of Shannon’s inequality states that for all real
k k
numbers p ,...,p ,q ,...q > 0, if p = 1 ≥ q then
1 k 1 k i=1 i i=1 i
P P
k k
−p log (p ) ≤ −p log (q )
i 2 i i 2 i
Xi=1 Xi=1
see [1] for a proof. As a consequence, for a tuple a = (a ,a ,...,a ) with
1 2 n
n
a ,...,a ∈ A and real numbers q(a) > 0 (a ∈ A) with q(a ) ≤ 1 we
1 n i=1 i
have P
n n
−log (p (a )) ≤ −log (q(a )). (2)
2 a i 2 i
Xi=1 Xi=1
4
2.2 Trees, tree sources, and tree compressors
With T we denote the set of all binary trees. We identify T with the
set of terms that are built from the binary symbol f and the constant a.
Formally, T is the smallest set such that (i) a ∈ T and (ii) if t ,t ∈ T
1 2
then also f(t ,t ) ∈T. With |t| we denote the number of occurrences of the
1 2
constant a in t. This is the number of leaves of t. Let T = {t ∈ T ||t| = n}
n
for n ≥ 1. Thedepth d(t) of the tree t is recursively definedby d(a) = 0 and
d(f(t ,t )) = max{d(t ),d(t )}+1. Let Td = {t ∈ T |d(t) = d} for d∈ N.
1 2 1 2
Occasionally, we will consider a binary tree t as a graph with nodes and
edges in the usual way. Note that a tree t ∈ T has 2n−1 nodes in total: n
n
leaves and n−1 internal nodes. For a node v we write t[v] for the subtree
rooted at v in t.
A context is a binary tree t, where exactly one leaf is labelled with the
special symbol x (called the parameter); all other leaves are labelled with
a. For a context t we define |t| to be the number of a-labelled leaves of t
(which is the number of leaves of t minus 1). We denote with C the set of all
contexts and define C = {t ∈ C | |t| = n} for n ∈ N. For a tree or context
n
t ∈ T ∪C and a context s ∈ C, we denote by s(t) the tree or context which
results from s by replacing the parameter x by t. For example s = f(a,x)
and t = f(a,a) yields s(t)= f(a,f(a,a)). The depth d(t) of a context t ∈ C
is defined as the depth of the tree t(a).
A tree source is a pair ((Fi)i∈N,P) such that the following conditions
hold:
• F ⊆ T is non-empty and finite for every i ≥ 0,
i
• F ∩F = ∅fori6= j and F = T,i.e., thesetsF formapartition
i j i≥0 i i
of T, S
• P : T → [0,1] and P(t)= 1 for every i≥ 0, i.e., P restricted to
t∈Fi
Fi is a probabilityPdistribution.
In this paper, we consider only two cases for the partition (Fi)i∈N: either
F = T for all i ∈ N (note that there is no tree of size 0) or F = Ti for all
i i+1 i
i ∈ N. Tree source of the former (resp., latter) type are called leaf-centric
(resp., depth-centric).
A tree encoder is an injective mapping E : T → {0,1}∗ such that the
range E(T) is prefix-free, i.e., there do not exist t,t′ ∈ T with t 6= t′ such
thatE(t)isaprefixofE(t′). Wedefinetheworst-case redundancy ofE with
respect to the tree source S = ((Fi)i∈N,P) as the mapping i 7→ R(E,S,i)
(i ∈ N) with
1
R(E,S,i) = max ·(|E(t)|+log P(t))
t∈Fi,P(t)>0 |t| 2
The worst-case redundancy is also known as the maximal pointwise redun-
dancy.
5
2.3 Tree straight-line programs
We now introduce tree straight-line programs for binary trees. Let V be a
finite ranked alphabet, where each symbol A ∈ V has an associated rank
0 or 1. The elements of V are called nonterminals. We assume that V
contains at least one element of rank 0 and that V is disjoint from the set
{f,a,x}, which are the labels used for binary trees and contexts. We use
V (respectively, V ) for the set of nonterminals of rank 0 (resp. of rank
0 1
1). The idea is that nonterminals from V (respectively, V ) derive to trees
0 1
from T (respectively, contexts from C). We denote by T the set of trees
V
over {f,a}∪V, i.e. each node in a tree t ∈ T is labelled with a symbol
V
from {f,a}∪V and the number of children of a node corresponds to the
rank of its label. With C we denote the corresponding set of all contexts,
V
i.e., the set of trees over {f,a,x}∪V, where the parameter symbol x occurs
exactly once and at a leaf position. Formally, we have T ⊂ T and C ⊂ C .
V V
A tree straight-line program G, or short TSLP, is a tuple (V,A ,r), where
0
A ∈ V isthestartnonterminalandr : V → (T ∪C )isthefunctionwhich
0 0 V V
assigns each nonterminal its unique right-hand side. It is required that if
A ∈ V (respectively, A ∈ V ), then r(A) ∈ T (respectively, r(A) ∈ C ).
0 1 V V
Furthermore, the binary relation {(A,B) ∈ V × V | B is a label in r(A)}
needs to be acyclic. These conditions ensure that exactly one tree is derived
from the start nonterminal A by using the rewrite rules A → r(A) for
0
A ∈ V. To define this formally, we define val (t) ∈ T for t ∈ T and
G V
val (t) ∈ C for t ∈ C inductively by the following rules:
G V
• val (a) = a and val (x) = x,
G G
• val (f(t ,t )) = f(val (t ),val (t )) for t ,t ∈ T ∪C (and t ∈ T
G 1 2 G 1 G 2 1 2 V V 1 V
or t ∈ T since there is at most one parameter in f(t ,t )),
2 V 1 2
• val (A) =val (r(A)) for A ∈ V ,
G G 0
• val (A(s)) = val (r(A))(val (s)) for A ∈ V , s ∈ T ∪C (note that
G G G 1 V V
val (r(A)) is a context t′, so we can built t′(val (s))).
G G
The tree defined by G is val(G) = val (A ) ∈ T. Moreover, for A ∈ V we
G 0 1
also write val (A) for val (A(x)).
G G
Example1. LetG = ({A ,A ,A },A ,r)beaTSLPwithA ,A ∈ V ,A ∈
0 1 2 0 0 1 0 2
V and
1
r(A ) = f(A ,A (a)), r(A ) = A (A (a)), r(A )= f(x,a).
0 1 2 1 2 2 2
Wegetval (A ) = f(x,a), val (A ) = f(f(a,a),a)and val(G) = val (A ) =
G 2 G 1 G 0
f(f(f(a,a),a),f(a,a)).
In this paper, we will consider two classes of syntactically restricted
TSLPs: (i) DAGs (directed acyclic graphs) and (ii) TSLPs in normal form.
Let us start with the former; normal form TSLPs will be introduced in
Section 4.1.
6
3 Tree compression with DAGs
In this section we sharpen some of the results from [17], where universal
source coding of binary trees using minimal DAGs (directed acyclic graphs)
is investigated. In [17], only bounds on the average redundancy for certain
classes of tree sources were shown. Here we extend these bounds (for the
same classes of tree sources) to the worst-case redundancy.
3.1 Directed acyclic graphs (DAGs)
A DAG is a TSLP D = (V,A ,r) such that V = {A ,A ,...,A } for
0 0 1 n−1
some n ∈ N, n ≥ 1, V = V (i.e., all nonterminals have rank 0), and
0
for every A ∈ V, the right-hand side r(A ) is of the form f(α ,α ) with
i i 1 2
α ,α ∈ {a,A ,...,A }. Note that a TSLP of this form generates a
1 2 i+1 n−1
tree with at least two leaves. In order to include the tree a with a single
leaf, we also allow the TSLP G = ({A },A ,A 7→ a). We define the size
a 0 0 0
of a DAG as |D|= n+1.
In contrast to general TSLPs, every binary tree t has a unique (up to
renaming of nonterminals) minimal DAG D , whose size is the number of
t
different (pairwise non-isomorphic) subtrees of t. The idea is to introduce
for every subtree f(t ,t ) of size at least two a nonterminal A with r(A ) =
1 2 i i
f(α ,α ), where α = a if t = a and α is the nonterminal corresponding to
1 2 i i i
the subtree t if |t | ≥ 2 (i = 1,2). We will only use this minimal DAG D
i i t
in the sequel.
Example 2. Consider the tree t = f(f(f(···f(a,a),···a),a),a), where
n
f occurs n times. We have |t | = n + 1. The minimal DAG of t is
n n
({A ,A ,...,A },A ,r ), where r (A ) = f(A ,a) for 0 ≤ i ≤ n−2
0 1 n−1 0 n n i i+1
and r (A ) = f(a,a) and its size is n+1.
n n−1
The above example shows that in the worst-case, the size of the minimal
DAG is not smaller than the size of the tree.
3.2 Universal source coding with DAGs
The following condition on a tree source was introduced in [17], where it is
called thedominationproperty(later, wewillintroduceastrongdomination
property): Let((Fi)i∈N,P)beatreesourceasdefinedinSection2.2. Wesay
that((Fi)i∈N,P)hastheweak domination property ifthereexists amapping
λ : T → R with the following properties:
>0
(i) λ(t)≥ P(t) for every t ∈T.
(ii) λ(f(s,t))≤ λ(s)·λ(t) for all s,t ∈ T
(iii) There are constants c ,c such that λ(t) ≤ c ·nc2 for all n ≥ 1.
1 2 t∈Tn 1
P
7
In [17], the authors define a binary encoding B(D ) ∈ {0,1}∗, such that
t
B(Dt)isnotaprefixofB(Dt′)forallbinarytreest,t′witht 6= t′. Theprecise
definitionofB(D )isnotimportantforus;allweneedisthefollowingbound
t
from [17, Theorem 2], where E : T → {0,1}∗ is the tree encoder with
dag
E (t) = B(D ).
dag t
Lemma 1. Assume that ((Fi)i∈N,P) has the weak domination property. Let
t ∈ T with n ≥ 2 and P(t) > 0, and let D be the minimal DAG for t. We
n t
have
1
·(|E (t)|+log (P(t))) ≤ O(|D |/n)+O(|D |/n·log (n/|D |)).
n dag 2 t t 2 t
Thisboundisusedin[17]toshowthatforcertain leaf-centricanddepth-
centric tree sources the encoding E is universal in the sense that the
dag
average redundancy converges to zero. Here, we want to show that for the
same tree sources already the worst-case redundancy converges to zero. Let
us first define the specific classes of tree sources studied in [17].
3.3 Leaf-centric binary tree sources
We recall the definition of a natural class of leaf-centric tree sources from
[17]: Let Σ be the set of all functions σ : (N\{0})×(N\{0}) → [0,1]
leaf
such that for all n ≥ 2:
σ(i,j) = 1. (3)
i,j≥1X,i+j=n
For σ ∈ Σ we define P : T → [0,1] inductively by:
leaf σ
P (a) = 1, (4)
σ
P (f(s,t)) = σ(|s|,|t|)·P (s)·P (t). (5)
σ σ σ
We have P (t) = 1 and thus ((T ) ,P ) is a leaf-centric tree source.
t∈Tn σ i i≥1 σ
The followPing result is implicitly shown in [17] (see also the proof of Theo-
rem 3).
Lemma 2. For every σ ∈ Σ , the leaf-centric tree source ((T ) ,P ) has
leaf i i≥1 σ
the weak domination property.
Wesaythatamappingσ ∈ Σ isleaf-balanced ifthereexistsaconstant
leaf
c such that for all (i,j) ∈ (N\{0})×(N\{0}) with σ(i,j) > 0 we have
i+j
≤ c.
min{i,j}
In[17]itisshownthatforaleaf-balancedσ ∈ Σ ,thetreesource((T ) ,P )
leaf i i≥1 σ
has the so called representation ratio negligibility property. This means
that the average compression ratio achieved by the minimal DAG (formally,
P (t)·|D |/n) converges to zero for n → ∞. Using a result from [5],
t∈Tn σ t
wPe show the following stronger property.
8
Lemma 3. For every leaf-balanced mapping σ ∈∈ Σ , there exists a con-
leaf
stant α such that for every binary tree t ∈ T with P (t) > 0 we have
n σ
|D |≤ α·n/log n.
t 2
Proof. In [5] the authors introduce the notion of β-balanced trees. Let v be
a non-leaf nodein t ∈ T, i.e. t[v] = f(t ,t ) for binary trees t and t . Then
1 2 1 2
v is β-balanced if |t |≤ β·|t | and |t | ≤ β·|t |. The tree t is β-balanced if
1 2 2 1
for all non-leaf nodes u and v in t such that u is a child node of v, at least
oneof the nodesu, v is β-balanced. Itis shown in[5]that for every constant
β thereexists aconstant α(dependingonly onβ)suchthattheminimaldag
D of a β-balanced tree t ∈ T has size at most α·n/log n. It follows that
t n 2
we only need to show that a tree t ∈ T with P (t) > 0 is β-balanced for a
n σ
constant β. Since the mapping σ is leaf-balanced, every subtree f(t ,t ) of
1 2
t satisfies |t |+|t | ≤ c·min{|t |,|t |}, where c ≥ 1 is a constant. Without
1 2 1 2
loss of generality assume that |t | ≤ |t |. We get |t |≤ c·|t | and
1 2 1 2
|t | ≤ |t |+|t |≤ c·min{|t |,|t |} = c·|t |,
2 1 2 1 2 1
which shows that t is c-balanced.
Corollary 1. Let σ ∈ Σ be leaf-balanced, and let S = ((T ) ,P ) be
leaf i i≥1 σ
the corresponding leaf-centric tree source. Then, we have R(E ,S,i) ≤
dag
O(loglogi/logi).
Proof. LetαbetheconstantfromLemma3. Lett ∈ T suchthatP (t) > 0.
n σ
Lemma 3 implies |D |≤ α·n/log n. With Lemma 1 and 2 we get
t 2
1
·(|E (t)|+log (P (t))) ≤ O(|D |/n)+O(|D |/n·log (n/|D |))
n dag 2 σ t t 2 t
≤ O(loglogn/logn)
This proves the corollary.
3.4 Depth-centric binary tree sources
We recall the definition of a natural class of depth-centric tree sources
from [17]: Let Σ be the set of all mappings σ : N × N → [0,1] such
depth
that for all n ≥ 1:
σ(i,j) = 1. (6)
i,j≥0,mXax(i,j)=n−1
For σ ∈ Σ , we define P : T → [0,1] by
depth σ
P (a) = 1, (7)
σ
P (f(s,t)) = σ(d(s),d(t))·P (s)·P (t). (8)
σ σ σ
We have P (t) = 1 and thus ((Ti) ,P ) is a depth-centric tree
t∈Tn σ i≥0 σ
source. P
The following result is again implicitly shown in [17] and can also be
found as a part of the proof of Theorem 4.
9
Lemma 4. For every σ ∈ Σ , the depth-centric tree source ((T ) ,P )
depth i i≥1 σ
has the weak domination property.
Wesaythemappingσ ∈ Σ isdepth-balancedifthereexistsaconstant
depth
c such that for all (i,j) ∈ N×N with σ(i,j) > 0 we have |i−j| ≤ c. In [17],
the authors define a condition on σ that is slightly stronger than depth-
balancedness, and show that for every such σ, the tree source ((Ti) ,P )
i≥0 σ
has the representation ratio negligibility property. Similarly to Lemma 3,
we will show an even stronger property. To do so, we introduce β-depth-
balanced trees for β ∈ N. A tree t is called β-depth-balanced if for each
subtree f(t ,t ) of t we have |d(t ) − d(t )| ≤ β. Note that for a depth-
1 2 1 2
balanced mapping σ ∈ Σ , there is a constant β such that every tree t
depth
with P (t) > 0 is β-depth-balanced. We will use the following lemma:
σ
Lemma 5. Let β ∈ N and c = 1+1/(1+β) (thus, 1 < c ≤ 2). For every
β-depth-balanced binary tree t, we have |t| ≥ cd(t).
Proof. We prove the lemma by induction on d(t). For the only tree t = a
of depth d(t) = 0, we have |t| = 1 = c0. Consider now a β-depth-balanced
tree t = f(t ,t ) of depth d(t) > 0. We assume d(t ) ≥ d(t ), the other case
1 2 1 2
is symmetric. Since t is β-depth-balanced, it follows that d(t )≥ d(t )−β.
2 1
To estimate the size |t| = |t |+|t |, we apply the induction hypothesis to t
1 2 1
and t , which yields
2
|t|= |t |+|t |≥ cd(t1)+cd(t2) ≥ cd(t1)+cd(t1)−β = cd(t1)· 1+c−β .
1 2
(cid:16) (cid:17)
Since d(t )+1 = d(t), it only remains to show that 1+c−β ≥ c, which can
1
be easily done by induction on β ∈ N.
Lemma 5 together with results from [5, 6] implies:
Lemma 6. For every depth-balanced mapping σ ∈ Σ there exists a
depth
constant α such that for every binary tree t ∈ T with P (t) > 0 we have
n σ
|D |≤ α·n·log (log n)/log n.
t 2 2 2
Proof. If P (t) > 0, then there exists a constant β such that t and each
σ
of its subtrees is β-depth-balanced. By Lemma 5 this implies that every
subtree t′ has depth at most c·log |t′| for a constant c that only depends
2
on σ. By [5, Theorem 12] (which was implicitly shown in [6]) it follows that
there exists a constant α (again, only dependent on σ) such that |D | ≤
t
α·n·log (log n)/log n.
2 2 2
Corollary 2. Let σ ∈ Σ be a depth-balanced mapping and let S =
depth
((Ti) ,P ) be the corresponding leaf-centric tree source. Then, we have
i≥0 σ
R(E ,S,i) ≤O((loglogi)2/logi).
dag
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