  A centroid node in a tree is a node for which the sum of the distances to all
other nodes attains its minimum, or equivalently a node with the property that
none of its branches contains more than half of the other nodes. We generalise
some known results regarding the behaviour of centroid nodes in random
recursive trees (due to Moon) to the class of very simple increasing trees,
which also includes the families of plane-oriented and $d$-ary increasing
trees. In particular, we derive limits of distributions and moments for the
depth and label of the centroid node nearest to the root, as well as for the
size of the subtree rooted at this node.
