The Colless index is a measure of balance used in phylogeny to study the shape of phylogenetic trees. In this paper, we study a more general class of objects, which we call mobiles, where leaves of a full binary tree have integer weights. We extend the Colless index to the more general problem of measuring balance in mobiles. We give a lower bound on the Colless index in the case of unit weights on n leaves, and give two classes of trees for which this bound is tight. We then turn to the more general case of mobiles, where we are given a list of integer weights and are asked to find a mobile of minimal Colless index. We identify instances of this problem that admit a polynomial-time algorithm, and show that in general it is in the parameterized class XP. The general problem is not known to be in P nor NP hard, a situation similar to the case of finding Huffman codes where the costs of the letters are not all equal. We give an integer linear program for both of these problems, that can easily be adapted to other Huffman-like problems with different cost evaluation functions.