Number the nodes as follows:
A valid solution is to check:
To see that this works, first note that the fox can only go from a hole in an even-numbered level of the tree to a hole in an odd-numbered level of the tree and vice-versa. If the fox starts in an odd level, the sequence 4,2,5,2,1,3,6,3,7 guarantees the hunter will catch it. This works because the hunter alternately looks at a hole at an odd level and then immediately checks the only possible remaining entrance to it, making sure the fox can never return to that hole:
If the fox has not been caught after this sequence, then you know it is in an even level of the tree. For the next day, it must have moved into a hole in an odd level, so we can use the same sequence again to seal its fate.