I don't recommend eating apple seeds, and it is not because a tree will grow in your stomach! Apple seeds, like many fruit seeds, contain compounds called cyanogenic glycosides, such as amygdalin, which act as part of the plant's natural defense mechanism. Cyanogenic glycosides are naturally occurring cyanide/sugar compounds that can release small amounts of toxic cyanide when their sugar groups are removed.
Prune and peach pits, which are well studied in the research world, have far greater concentrations of cyanogenic glycosides than apple seeds; however, the lower levels in apple seeds could still pose problems in terms of stomachache or food poisoning, and a few seeds chewed by a small child could result in poisoning. Although the severity of the problem caused by ingesting apple seeds would depend upon many factors, including a person's health, I don't think that the potential risk here is worth taking.
For more information on this topic, please see:
Holzbecher MD, Moss MA, Ellenberger HA. The Cyanide Content of Laetrile Preparations, Apricot, Peach and Apple Seeds. J Toxicol Clin Toxicol. 1984;22(4):341-7.
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