Leftist Trees:Weight-Biased Leftist Trees

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Weight-Biased Leftist Trees

Definition

We arrive at another variety of leftist tree by considering the number of nodes in a subtree, rather than the length of a shortest root to external node path. Define the weight w(x) of node x to be the number of internal nodes in the subtree with root x. Notice that if x is an external node, its weight is 0. If x is an internal node, its weight is 1 more than the sum of the weights of its children. The weights of the nodes of the binary tree of Figure 5.1(a) appear in Figure 5.1(d)

DEFINITION 5.4 [Cho and Sahni [2]] A binary tree is a weight-biased leftist tree (WBLT) iff at every internal node the w value of the left child is greater than or equal to the w value of the right child. A max (min) WBLT is a max (min) tree that is also a WBLT.

Note that the binary tree of Figure 5.1(a) is not a WBLT. However, all three of the binary trees of Figure 5.2 are WBLTs.

THEOREM 5.2 Let x be any internal node of a weight-biased leftist tree. The length,

rightmost(x), of the right-most path from x to an external node satisfies

rightmost(x) ≤ log2(w(x) + 1).

Proof The proof is by induction on w(x). When w(x) = 1, rightmost(x) = 1 and log2(w(x) + 1) = log2 2 = 1. For the induction hypothesis, assume that rightmost(x) ≤

log2(w(x)+ 1) whenever w(x) < n. Let RightChild(x) denote the right child of x (note that this right child may be an external node). When w(x) = n, w(RightChild(x)) ≤ (n − 1)/2 and rightmost(x) = 1 + rightmost(RightChild(x)) ≤ 1+ log2((n − 1)/2 + 1) = 1 + log2(n + 1) − 1 = log2(n + 1).

Max WBLT Operations

Insert, delete max, and initialization are analogous to the corresponding max HBLT operation. However, the meld operation can be done in a single top-to-bottom pass (recall that the meld operation of an HBLT performs a top-to-bottom pass as the recursion unfolds and then a bottom-to-top pass in which subtrees are possibly swapped and s-values updated). A single-pass meld is possible for WBLTs because we can determine the w values on the way down and so, on the way down, we can update w-values and swap subtrees as necessary. For HBLTs, a node’s new s value cannot be determined on the way down the tree.

Since the meld operation of a WBLT may be implemented using a single top-to-bottom pass, inserts and deletes also use a single top-to-bottom pass. Because of this, inserts and deletes are faster, by a constant factor, in a WBLT than in an HBLT [2]. However, from a WBLT, we cannot delete the element in an arbitrarily located node, theN ode, in O(log n) time. This is because theN ode may have O(n) ancestors whose w value is to be updated. So, WBLTs are not suitable for mergeable double-ended priority queue applications [3, 8].

C++ and Java codes for HBLTs and WBLTs may be obtained from [9] and [10], respectively.

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