On the distribution of binary search trees under the random permutation model

Multiway trees, also known as m–ary search trees, are data structures generalising binary search trees. A common probability model for analysing the behaviour of these structures is the random permutation model. The probability mass function Q on the set of m–ary search trees under the random permutation model is the distribution induced by sequentially inserting the records of a uniformly random permutation into an initially empty m–ary search tree. We study some basic properties of the functional Q, which serves as a measure of the ‘shape’ of the tree. In particular, we determine exact and asymptotic expressions for the maximum and minimum values of Q and identify and count the trees achieving those values.

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An almost sure result for path lengths in binary search trees

Advances in Applied Probability ◽

10.1017/s0001867800012295 ◽

2003 ◽

Vol 35 (02) ◽

pp. 363-376

Author(s):

F. M. Dekking ◽

L. E. Meester

Keyword(s):

Path Length ◽

Random Variable ◽

Random Permutation ◽

Search Tree ◽

Binary Search ◽

Binary Search Tree ◽

Binary Search Trees ◽

Search Trees ◽

Total Path Length ◽

Path Lengths

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variableZ. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜mof finite binary trees with at mostmnodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector timesZ.

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An almost sure result for path lengths in binary search trees

Advances in Applied Probability ◽

10.1239/aap/1051201652 ◽

2003 ◽

Vol 35 (2) ◽

pp. 363-376 ◽

Cited By ~ 1

Author(s):

F. M. Dekking ◽

L. E. Meester

Keyword(s):

Path Length ◽

Random Variable ◽

Random Permutation ◽

Search Tree ◽

Binary Search ◽

Binary Search Tree ◽

Binary Search Trees ◽

Search Trees ◽

Total Path Length ◽

Path Lengths

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variable Z. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜m of finite binary trees with at most m nodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector times Z.

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Almost sure asymptotics for the random binary search tree

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2775 ◽

2010 ◽

Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽

Author(s):

Matthew Roberts

Keyword(s):

Random Permutation ◽

Search Tree ◽

Binary Search ◽

Saturation Level ◽

Binary Search Tree ◽

International Audience ◽

Permutation Model ◽

Random Permutation Model

International audience We consider a (random permutation model) binary search tree with $n$ nodes and give asymptotics on the $\log$ $\log$ scale for the height $H_n$ and saturation level $h_n$ of the tree as $n \to \infty$, both almost surely and in probability. We then consider the number $F_n$ of particles at level $H_n$ at time $n$, and show that $F_n$ is unbounded almost surely.

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