Small tree, old as it happens (5)

I believe the answer is:

olive

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'small tree' is the definition.
'olive' can be an answer for 'tree' (I've seen this before). I'm not certain of the 'small' bit.

'old as it happens' is the wordplay.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'as it happens' becomes 'live' (as in live television).
'o'+'live'='OLIVE'

Can you help me to learn more?

(Other definitions for olive that I've seen before include "Fruit; Popeye's girlfriend" , "shades of green" , "Stoned fruit" , "Greyish-green colour" , "yellowish green" .)

I've seen this clue in The Telegraph.

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