B Proof of Lemma 4.3

(1) Suppose that w belongs to a C d , d r. The total number of leaves in C 0 ; : : :; C d?1 is P d?1 j=0 2 j = 2 d ? 1. Therefore, 2 d ? 1 < i. Since w belongs to C d , i P d j=0 2 j = 2 d+1 ? 1. Thus, 2 d i < 2 d+1 , implying that d = blog ic. Consider the path from the root of T r to w. The path has length 2d + 1: d edges on the path from the root of T r to v d , one edge connecting v d and the root of C d , and d edges on the path from the root of C d to w. Since d = blog w i c, the length of the path is 2blog ic + 1. (2) Deene a binary string b1 : : :2d + 1], which represents the path from the root of T r to w, as follows: b1] = 1; for any 1 j < 2d + 1, bj + 1] = 0, if the (j + 1)th vertex on the path is the left child of its parent; and bj + 1] = 1, otherwise. The numbering of the vertices in T r implies that the number of w is represented by the binary string b1 : : :2d + 1]. The d leftmost bits of b, corresponding to the path from the root of the tree to vertex v d are 11 : : :1, since all the vertices on the path are right children (see Figure 3(b)). The (d+1)th bit of b is 0, since the root of C d is the left child of v d. It remains to calculate the d rightmost bits of b, corresponding to the path from the root of C d to w. The total number of leaves in C 0 ; : : :; C d?1 is P d?1 j=0 2 j = 2 d ?1. Since w is the ith leaf in T r , it is the (i ?(2 d ?1))th leaf in C d , counting from 1. Therefore, the d bits corresponding to the path from the root of C d to w are the binary representation of the number (i ? (2 d ? 1)) …