I had to check out what a lime tree was. I thought they were the citrus fruit for putting in rum and coke. Nope, fooled by common names again. Below is taken from wikipedia. In case you didn't know.
Lime is an altered form of Middle English lind, in the 16th century also line, from Old English feminine lind or linde, Proto-Germanic *lendā, cognate to Latin lentus "flexible" and Sanskrit latā "liana". Within Germanic languages, English lithe, German lind "lenient, yielding" are from the same root.
Linden was originally the adjective, "made from lime-wood" (equivalent to "wooden"), from the late 16th century "linden" was also used as a noun, probably influenced by translations of German romance, as an adoption of Linden, the plural of German Linde.[1] Neither the name nor the tree is related to the citrus fruit called "lime" (Citrus aurantifolia, family Rutaceae). Another common name used in North America is basswood, derived from bast, the name for the inner bark (see Uses, below). In the US, the name "lime" is used only for the citrus tree. Teil is an old name for the lime tree.