(For those of you who saw my prior post: I misunderstood the OP's reply, so I have edited. Use this one instead.)
I also have doubts about the correct model, and I do not have enough information about your research questions to do much speculation.
The dilemma of collapsing observations within each year to months is that a monthly basis is arguably arbitrary. How do you justify collapsing to a month level? How do you justify the date cutpoints that determine "month"? Trees do not recognize the human concept of "June" versus "July", for example. Is "June" in year 1 equivalent to "June" in year 2 equivalent to "June" in year 3? In the field, this is not usually the case because of annual variability in weather. Should year be a factor that is separate from month?
When in doubt, go back to your research questions. Why did you measure over 3 years? What did you expect to learn from the temporal observations within each year? What did you expect to learn from the observations over multiple years? What aspect of these temporal observations is fundamental to your research questions? For example, perhaps you are primarily interested in annual growth and care less about the growth within each year.
34 observations per tree (= 3 x 11 plus an extra? = 3 x 12 minus 2?) is still a lot for an interpretable ANOVA. I'd give it more thought. If you have statistical consultation available to you at your institution, take advantage of it, particularly if the consultant has some familiarity with your discipline.
I hope this helps.
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