... vertices, v1 and v2, with thecentres of stars. We shall write T = C(a, b, c, d, e), if P has length a + c+ e, and v1 and v2are the (a + 1)th and (a + c + 1)th vertex on P , and have b, resp. ... k, and the set S := V (G) \ L.Hence, the Loebl–Koml´os–S´os conjecture is true for all trees of diameter at most 3. Barr and Johansson [2], and independently Sun [11], proved the conjecture for ... d, neighbours outside P .Therefore, if a, e > 0, then C(a, b, c, d, e) has b + d + 2 leaves.We call P the body, and v1 and v2the joints of the caterpillar. For illustration, seeFigure...