I've got a house which apparently used to have a number of flower beds all around. We have a few flowers and decorative plants growing in them, but we also get a regular population of Walnut trees. I cannot pull them, so I've been pruning them back brutally. I know there is energy stored in the roots, because the darn things keep popping back up.
How many times should I expect to have to cut these things down (just above ground level) before they stay down? Is there a particular time of year that will work better?
Just keep pruning...eventually they will get the message and stop growing. For those who are not afraid of chemical control, the "glove of death" with round-up works well. Rubber or vinyl glove with a cloth glove over that sprayed with round-up and hand applied to the saplings. This ensures only the walnut trees get poisoned. Glyphosate will not move to other plants.