The if the tree is on his property, then the landlord would be responsible for the bill. However in many cases if you talk to the water company, and explain what happened, they can often reduce the amount of the bill (or waive it entirely) if it's due to something like that.
I think neither you nor your landlord is responsible. Your water services provider is supposed to take charge. If it is at a position where it will read on your meter, the landlord might be the one to cater for that. You pay for what you have used. Some instances would also depend on consensus like if you had already used some water and it is towards the end of the period after which you pay, you can agree to pay your regular amount.