Answer: The repeated stress is undoubtedly not beneficial to the tree. Although river birches are more tolerant of dry soil (once established) than the typical white birch, it does require adequate soil moisture during the establishement phase while it sends roots out beyond the original root ball and into the surrounding native soil.
Unfortunately, the drought we have experienced during the past two years has been very dificult for many of our woody plants, both old and well established and the newly planted. Each time you water, the drier surrounding soil will wick that moisture away. This means you do need to be very careful about watering.
It is possible that the watering you have been doing has not really been soaking the entire rootball plus some of the surrounding soil -- roots will not grow into dry soil so you need to make sure that both are moist. Also, since birches are somewhat shallow rooted, you would need to water more often to keep the root zone moist.
There is no rule of thumb I can give you on watering, it is a matter of trial and error to some extent. You should water, wait a few hours, then dig down and see how effective your watering was (or wasn't) since it can be surprising. Then, in order to tell if and when you need to water again, dig down into the soil with your finger and see. Your goal is to keep the soil evenly moist, but not sopping wet. (Birches are very tolerant of wet soil however so overwatering occasionally would not be harmful, in fact they tend to do well planted in a soil that tends to stay moister naturally.) YOu are supplementing natural rainfall and snow melt, so you would need to take that into account. Hot, dry and/or windy weather will also be more drying than cooler weather.
Finally, the larger the diameter of the tree trunk at planting, the more years it will take to become established. Typically trees need careful watering the first year or two, then watering in dry spells for the next several years. In severe drought conditions, even established trees may need supplemental watering, so this is not really so much to ask when you think of the decades of enjoyment a tree can provide.
I'm sorry you have been having such disappointment with your tree. There is always the possibility that some other factor besides water is at work, such as a pest problem or perhaps the soil where the tree is planted is incompatible or some other physical problem such as underlying rock or other impediment to growth is at work. You might want to consult with the nursery where you purchased the tree or with your county extension or possibly with a professionally trained and certified arborist to see if there is some combination of factors working against you that could be fixed. Good luck with your tree.
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