Let T be a tree of order n. For any edge labeling f : E → {1, 2, 3, ...} the weight of a path P is the sum of the labels of the edges of P and is denoted by w(P). If the weights of the n C 2 paths in T are exactly 1, 2,..., n C 2 , then f is called a Leech labeling and a tree which admits a Leech labeling is called a Leech tree. In this paper we determine all Leech trees having diameter three. We also prove that the tree obtained from the path P n = (v 1 , v 2 , ..., v n) by attaching a pendent vertex at v n−1 is not a Leech tree for all n ≥ 4.

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