Tree Flow

Djenka has a rooted tree with $N$ nodes (numbered $1$ through $N$) and $N-1$ edges; node $1$ is the root. For each $i$ ($1 \le i \le N-1$), the $i$-th edge connects node $i$ with its parent node $p_i$ and it has capacity $c_i$. Each node can store an arbitrary non-negative amount of water. The capacity of each edge denotes the maximum amount of water which can flow across this edge in one second