Even for cherry blossoms, they put off pollen, so you could probably do something similar. It'd be tricky to get a fine enough resolution (I'd be impressed if you could even get estimates as specific as month-level). Though the millenium of data would be awesome for verifying the proxy.
> Cores are drilled with hand augers (for shallow holes) or powered drills; they can reach depths of over 3.2 km, and contain ice up to 800,000 years old.
> The physical properties of the ice and of material trapped in it can be used to reconstruct the climate over the age range of the core. The proportions of different oxygen and hydrogen isotopes provide information about ancient temperatures, and the air trapped in tiny bubbles can be analysed to determine the level of atmospheric gases such as carbon dioxide.
https://en.wikipedia.org/wiki/Ice_core (edited for SI units preference)