We always have lots of baby birch trees appearing all over the garden, especially anywhere with a gravel mulch. Total pain! We have dug up lots to transplant into our tree areas, but there is a limit and it looks as if we'll reach it next year
A mast year, presumably named after beech mast ie seeds, happens for different tree types in different years as I understand it. It's when that tree species produces an overabundance of seeds in an apparent effort to swamp the ground with their seeds, presumably to increase the chance of more of their species taking root.
It was either last year or the year before that Sycamores had a mast year and social media was full of people asking for ID of seedlings found throughout their gardens. Just as I'll be finding next year.
Your dogs helping with removing the seedlings is brilliant
It reminded me of the year our Heinz 57 wonderful dog Calli helped us to remove the old stalks of brassicas such as Purple Sprouting Broccoli and brussels Sprouts. She was really good at it, but of course the following year she was found joyously ripping out our new crop 
How to explain?