Granpa can be heard singing a song called "Wild Mountain Thyme" (A.K.A. "Will You Go Lassie, Go") several times throughout the movie. While the song is a variant of "The Braes of Balquhidder" by Robert Tannahill (1774-1810), the lyrics and the tune sung by Granpa are for the "Wild Mountain Thyme" version, which was written and published by Francis McPeake (from Belfast, Northern Ireland) in 1957, approximately 22 years after the movie takes place.
When Little Tree's grandmother is serving the meal, there's a large spoon to the right of her plate. Then there's a close-up shot of Little Tree as she sets the two bowls on her and Little Tree's plates and then fetches the dictionary and her glasses, and a wide shot of the whole table again. But now the spoon has vanished. When she finishes reading to him out of the dictionary, there's another wide shot and the spoon has reappeared.
When Little Tree is fetching wood chips at Jenkins' store, there's an axe stuck in the stump. When the girl first comes over to talk with him, the axe handle is down at about an angle of 45 degrees. After she touches his moccasins and they talk about the fact that he's an Indian, there's a wide shot and the axe handle has been repositioned so that it's parallel to the ground.
When Little Tree arrives in the car at the Notch Gap Indian School, when he gets out of the car, a large reflector is visible reflected in the car door, as is the reflected outline of a crew member holding it.