This usually means a nest, with the young ones demanding food.
Keeping a close watch from the corners of my eyes (while continuing to weed) I could see the yellow front and black head of a Great Tit, hovering around the trunk of the tree and chip-chipping at me. At first I thought there must be a nest in the branches, but as I worked around the base of the tree, I realised that the cheeping was actually coming from within the tree trunk.
There are several holes in the trunk - as I said, it's an old tree - and I guessed that the nest must be inside the actual trunk. After a while, the parents bird became bolder (or possibly more desperate? I do hope not) and were flying right into the hole to feed the little ones, even though I was working just a few feet away.
They seemed to take quite some time to pass out the food inside the trunk, so I sneakily waited until the parent had gone inside, then tiptoed up to the trunk, camera-phone in hand and all ready.
Here it is: be patient, watch the hole, and don't blink:
The parent bird popped out of the hole, took one look at me, chirped in fright, and swerved away around the trunk.