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Leaves on sprout shoots like these aren't reliable for identifying Tilia to species - I don't know whether this is just because the differ from the "normal" leaves described in floras and keys, or whether they really aren't distinguishable. And, as you observe, they can be pretty big.
However, the combination of a simple, unlobed, leaf, with palmate venation at the base is sufficient to identify lime in this country.
The large leaf in the 3rd pic was actually plucked from overhanging branches out on the larger limbs. I would have mentioned this but I wasn't aware that the location of the leaf on the tree would matter much- as long as it was from the same tree.
Which details would be helpful for an attempt at IDing the species of this lime?
Back when I was less skilled in identification of trees I wrote a few words on the identification of limes.
Thanks for this, I might see if I can use it to form a potential ID for my lime.
Blimey, if you wrote that when you were less skilled, how incredibly good are you now?
Seriously, I must say that I particularly liked and admired your fourth paragraph, where every technical term is immediately clarified in simple language. (Read it, everyone, and learn!)
How to take close-ups with cheap phone and hand-lens:
Field Guides for Budding Botanists:
Lat/Lng: 55.0828, -6.06
OS grid ref: NW409840