[verb (dictionary form)] + たびに [noun] + の + たびに
たびに means "every time", "whenever": each time one thing happens, the other happens too. It follows a verb's dictionary form, or a noun with の. It is also written in kanji, どに度に.
The second half is something the speaker notices or feels each time: a memory, a change, a reaction. A plain daily routine does not take たびに; say it with まいあさ毎朝 or いつも.