Beneath tree that's heavenly (7)

I believe the answer is:

sublime

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'heavenly' is the definition.
(I've seen this in another clue)

'beneath tree' is the wordplay.
'beneath' becomes 'sub' (I can't justify this - if you can you should believe this answer much more).
'tree' becomes 'lime tree' (type of tree).
'sub'+'lime'='SUBLIME'

'that's' is the link.

Can you help me to learn more?

(Other definitions for sublime that I've seen before include "Heavenly" , "Of great excellence or beauty, awe-inspiring" , "Unparalleled, extreme" , "Exalted; awesome" , "Glorious; supreme" .)

I've seen this clue in The Guardian.

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