, 2016

We label the remaining points on the diagram as shown.

[[IMAGE0]]

There is exactly one path that the squirrel can take to get to each of $A$, $C$, $F$, $B$, $E$, and $J$. For example, to get to $F$ the squirrel must walk from $P$ to $A$ to $C$ to $F$. The number of paths that the squirrel can take to point $D$ is 2, since there is 1 path to each of $A$ and $B$, and to get to $D$, the squirrel must go through exactly one of $A$ or $B$. Similarly, the number of paths to $G$ is the sum of the number of paths to $C$ and to $D$ (that is, $1+2=3$), because for the squirrel to get to $G$, it must walk through exactly one of $C$ or $D$. Using this process, we add to the diagram the number of paths to reach each of $H$, $I$, $K$, and $L$. [[IMAGE1]] Finally, to get to $Q$, the squirrel must go through exactly one of $H$, $K$, or $L$, so the number of paths to $Q$ is $6+4+4=14$.