Hairy Search Trees

Random search trees have the property that their depth depends on the order in which they are built. They have to be balanced in order to obtain a more eecient storage-and-retrieval data structure. Balancing a search tree is time consuming. This explains the popularity of data structures which approximate a balanced tree but have lower amortised balancing costs, such as AVL trees, Fibonacci trees and 2-3 trees. The algorithms for maintaining these data structures eeciently are complex and hard to derive. This observation has led to insertion algorithms that perform local balancing around the newly inserted node, without backtracking on the search path. This is also called a fringe heuristic. The resulting class of trees is referred to as 1-locally balanced trees, in this note referred to as hairy trees. In this note a simple analysis of their behaviour is povided. 1 Hairy Trees Locally balanced search trees have been invented and analysed a long time ago ((PM85, GON83]), but they have not become as popular as unbalanced search trees or AVL-trees. In this note, we show how to obtain a simple form of locally balanced trees and to analyse their behaviour. These hairy trees are a class of search trees, characterized by: is hairy (t) 8 node v 2 v has single son s) s is leaf] The intuition behind this condition is that it prevents trees from having list-like substructures longer than two nodes (\bare twigs"). Some examples of hairy trees are presented in gure 1. 2 1 2 1 4 3 J J J 1 3 2 J J J Figure 1: Two hairy trees and a non-hairy one The class of hairy trees can be described by the following recursive deenition: 1. is hairy (") 2. If t is a singleton tree and x some key value, then: is hairy (t) is hairy (Tree (x; t; ")) is hairy (Tree (x; " ; t)). 3. If t 1 and t 2 are both non-empty hairy trees and x is some key value, then is hairy(Tree(x; t 1 ; t 2)). where " is the empty tree, and Tree (x; t 1 ; t 2) the constructor of trees. We will also overload this con-structor for singleton trees: Tree (x) = Tree (x; " ; "). The above inductive deenitions give us the opportunity to use structural induction in reasoning about hairy trees. Deenition 1 The function …