Lime trees are what make up the lime-walk (presumably a part of the manor’s gardens) and are the secret meeting place of the old man’s grandmother and Antonio. The lime-walk is mentioned twice:
First, after the Duchess’s fervent devotion begins and the maids are instructed by her to warn her if anyone approaches the chapel. The old man’s grandmother fears being found meeting with Antonio rather than doing Nencia’s bidding: “That it was her own fault I won’t deny, for she’d been down the lime-walk with Antonio when her aunt fancied her to be stitching in her chamber; and seeing a sudden light in Nencia’s window, she took fright lest her disobedience be found out, and ran up quickly through the laurel-grove to the house.”214
And the other after the old man’s grandmother overcomes her fright of hearing and seeing someone drop from the window of the chapel: “Well, my grandmother turned the matter over, and next time she met Antonio in the lime-walk (which, by reason of her fright, was not for some days) she laid before him what had happened; but to her surprise he only laughed and said, ‘You little simpleton, he wasn’t getting out of the window, he was trying to look in’; and not another word could she get from him.”215
213 Wharton 1901, p. 11
214 Wharton 1901, p. 11
215 Wharton 1901, p. 12
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Lime trees, also known as linden trees, are the tree of love216. It is no wonder then, that the courting Antonio and old man's grandmother meet by them for their after-hours flirtations.