Due to their inherent flexibility to be exercised at any time before expiration, American options present a valuation challenge that is more complex and costly compared to European options, which can only be exercised at expiration. This paper seeks to elucidate the premium of American options through a 2-period binomial tree model under the assumption that the product of downside and upside probabilities equals one. This work explores the price differential between American and European options across a spectrum of interest rates r and strike prices K. Furthermore, this work investigates the implications of dividends on American call option premiums by employing a 1-period binomial tree model. The inclusion of dividends adds a layer of complexity to the valuation process, necessitating the derivation of risk-neutral probabilities that account for dividends while maintaining no-arbitrage conditions. Our analysis reveals how variations in interest rates, strike prices, and the presence of dividends significantly influence American option premiums. By providing a detailed framework, our findings contribute to a deeper understanding of the pricing dynamics of American options, enabling better strategic decision-making for investors and financial practitioners alike. This comprehensive approach sheds light on the intricate relationships that govern option valuation in diverse market conditions.