A height balanced tree is a rooted binary tree T in which for every vertex v∈V(T), the difference b _T (v) between the heights of the subtrees, rooted at the left and right child of v is at most one. We show that a height-balanced tree T _h of height h is a subtree of the hypercube Q _h+1 of dimension h+1, if T _h contains a path P from its root to a leaf such that , for every non-leaf vertex v in P. A Fibonacci tree is a height balanced tree T _h of height h in which , for every non-leaf vertex. has f(h+2)−1 vertices where f(h+2) denotes the (h+2)th Fibonacci number. Since f(h)∼2^0.694h , it follows that if is a subtree of Q _n , then n is at least 0.694(h+2). We prove that is a subtree of the hypercube of dimension approximately 0.75h.