On the dimension of trees

We consider isometric embedding of trees into the infinite graph Zm whose vertices are the m-dimensional lattice points where two vertices a=(a1,a2,…,am) and b=(b1,b2,…,bm) are adjacent if and only if |ai-bi|⩽1 for 1⩽i⩽m. Linial, London, and Rabinovich have shown that this can be done with m⩽1.7095log2t, where t is the number of leaves. In this note, we sketch a proof that ⌈log2t⌉⩽m⩽⌈1.45log2t⌉.