This article studies sums over all compositions of an integer. We derive a
generating function for this quantity, and apply it to several special
functions, including various generalized Bernoulli numbers. We connect
composition sums with a recursive tree introduced by S.G. Woon and extended by
P. Fuchs under the name "general PI tree", in which an output sequence
{xn​} is associated to the input sequence {gn​} by summing over each
row of the tree built from {gn​}. Our link with the notion of compositions
allows to introduce a modification of Fuchs' tree that takes into account
nonlinear transforms of the generating function of the input sequence. We also
introduce the notion of \textit{generalized sums over compositions}, where we
look at composition sums over each part of a composition