The height of record‐biased trees

Abstract

Given a permutation σ$$ \sigma $$ , its corresponding binary search tree is obtained by recursively inserting the values σ(1),…,σ(n)$$ \sigma (1),\dots, \sigma (n) $$ into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In this article, we study the height of binary search trees drawn from the record‐biased model of permutations whose probability measure on the set of permutations is proportional to θrecord(σ)$$ {\theta}^{\mathrm{record}\left(\sigma \right)} $$ , where record(σ)=|{i∈[n]:∀jσ(j)}|$$ \mathrm{record}\left(\sigma \right)=\mid \left\{i\in \left[n\right]:\forall j\sigma (j)\right\}\mid $$ . We show that the height of a binary search tree built from a record‐biased permutation of size n$$ n $$ with parameter θ$$ \theta $$ is of order (1+oℙ(1))max{c∗logn,θlog(1+n/θ)}$$ \left(1+{o}_{\mathbb{P}}(1)\right)\max \left\{{c}^{\ast}\log n,\kern0.3em \theta \log \left(1+n/\theta \right)\right\} $$ , hence extending previous results of Devroye on the height or random binary search trees.