  We present a dynamic data structure for maintaining the persistent homology
of a time series of real numbers. The data structure supports local operations,
including the insertion and deletion of an item and the cutting and
concatenating of lists, each in time $O(\log n + k)$, in which $n$ counts the
critical items and $k$ the changes in the augmented persistence diagram. To
achieve this, we design a tailor-made tree structure with an unconventional
representation, referred to as banana tree, which may be useful in its own
right.
