Summary 1. The study of invasiveness typically emphasizes early successional life‐history traits in exotic plants, which enable the capture of high resources in disturbed environments and rapid growth. A key issue in invasion dynamics is whether such behaviours come at the expense of traits such as low‐light survivorship, which allow species to become more dominant later in succession. 2. We used maximum‐likelihood analysis to compare the growth and survivorship of two exotic trees, Ailanthus altissima and Acer platanoides, with nine dominant native tree species in closed‐canopy forests in Connecticut, USA. Growth was modelled as a function of light and survivorship as a function of recent growth; combining models yielded estimates of light‐dependent mortality. 3. The exotic species had strikingly high growth rates, exceeding all native species at light levels ≥ 10% full sun, and growing 2.6 times faster than the fastest‐growing native species at 80% full sun. At low‐light levels (< 3% full sun), however, growth rates of five native species exceeded both exotics. Exotic species survivorship (as a function of light‐driven growth) was strongly dependent on the degree of shading: at 1% full sun, the annual mortality rate of A. platanoides was 10% and A. altissima was 17%; only two native species had higher mortalities. However, at 5% full sun, A. platanoides’ mortality was < 1%, superior to all but three native species. Mortality of all species dropped to < 1% by 10% full sun, except A. altissima whose mortality remained high at c. 10%. 4. A life‐history trade‐off analysis (based on radial growth, height allometry and low‐light survivorship) shows a nearly linear trade‐off for most species. The native species and Ailanthus follow the common life‐history trade‐off of low‐light survivorship vs. high‐light growth. However, A. platanoides diverges from this trade‐off pattern by combining very high growth rates with moderately high shade tolerance. 5. Simulations with SORTIE‐ND (a forest dynamics model) indicate that poor survivorship of A. altissima will limit it to disturbed sites, whereas A. platanoides’ unusual combination of traits makes it invasive in both disturbed and undisturbed forests. Overall, native shade‐tolerant trees and slow stand dynamics make undisturbed forests highly resistant to invasion by exotic trees that are intolerant of shade. 6. Synthesis. This study showcases the importance of rapid growth in invasive plants, holding even for exotic tree species known to invade established forests. For A. altissima, high growth rates were accompanied by poor low‐light survivorship. A. platanoides departs from the general trade‐off pattern that exists among native species and A. altissima, and consequently it can be highly invasive in closed‐canopy forests.