As we dress the Torah in my congregation and many congregations I've been to, we sing (from the siddur's liturgy on putting away the Torah):
עֵץ חַיִּים הִיא לַמַּחֲזִיקִים בָּהּ, וְתוֹמְכֶיהָ מְאֻשָּׁר
דְּרָכֶיהָ דַרְכֵי נוֹעַם, וְכָל נְתִיבוֹתֶיהָ שָׁלוֹם
אֹרֶךְ יָמִים בִּימִינָהּ, בִּשְׂמֹאולָהּ עֹשֶׁר וְכָבוֹד
יְהוָה חָפֵץ, לְמַעַן צִדְקוֹ; יַגְדִּיל תּוֹרָה, וְיַאְדִּיר
and return to the siddur (Aichah 5:21):
חַשִׁבֵנוּ ה' אֵלֶי וְנָשׁוּבָה חַדֵּשׁ יָמֵינוּ כְּקֶדֶם
What minhag is it to insert the verses from Mishlei and Yeshayahu and why do we do it? Or, conversely, what minhag omits those verses and why do no siddurs make mention of them?
(NOTE: I may have the order wrong. Hashiveinu may come before Orech Yamim).
This is the tune I'm familiar with, though it only goes up to נְתִיבוֹתֶיהָ שָׁלוֹם. Neither of these tunes are remotely the tune that my shuls use, but both include Etz Chayyim Hi the two additional verses which implies to me that it's more than just a solitary composer's decision.