On embedding subclasses of height-balanced trees in hypercubes

A height-balanced tree is a rooted binary tree T in which for every vertex image, the heights of the subtrees, rooted at the left and right child of v, differ by at most one; this difference is called the balance factor of v. These trees are extensively used as data structures for sorting and searching. We embed several subclasses of height-balanced trees of height h in image under certain conditions. In particular, if a tree T is such that the balance factor of every vertex in the first three levels is arbitrary (0 or 1) and the balance factor of every other vertex is zero, then we prove that T is embeddable in its optimal hypercube with dilation 1 or 2 according to whether it is balanced or not.