Answer: Some of the smaller trees, called patio trees, will grow happily in the site you describe. Evergreen tree suggestions include Acacia, Citrus (lemon or lime), Cassia (gold medallion tree), Leptospermum (Australian tea tree), Pyrus kawakamii (evergreen pear) and Stenocarpus sinuatus (firewheel tree). Deciduous tree suggestions include Cercidiphyllum (katsura), Erythrina (coral tree), Malus (flowering crabapple), Laburnum (goldenchain) and Vitex (chaste tree). Hope one of these suggestions appeals to you!
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