Final answer:
Immigration was the key factor in the population growth of cities like New York during the mid-nineteenth century, as technological advances and the Second Industrial Revolution created a high demand for labor in urban centers.
Step-by-step explanation:
The critical factor in the ballooning populations of cities like New York in the mid-nineteenth century was immigration. Technological and economic factors like the development of the steam engine, which allowed factories to locate near urban centers, and the subsequent need for a large workforce contributed to rapid urbanization. Coupled with the fact that America was entering the Second Industrial Revolution, urban centers provided a concentration of jobs that attracted both domestic migrants from rural areas and a significant influx of immigrants, particularly from eastern and southern Europe, who were seeking better opportunities and sometimes fleeing famine or religious persecution. This massive influx of job-seekers transformed the cities into bustling hubs of diverse populations and industrial activity. Additionally, improvements in sanitation, health standards, and urban infrastructure, including transportation systems like elevated railroads, helped accommodate and support the growing populations and further stimulate urban growth.